Solve -3x^2 + 6x = -2
In the realm of mathematics, quadratic equations stand as fundamental tools for modeling a vast array of real-world phenomena, from the trajectory of projectiles to the optimization of engineering designs. A quadratic equation, characterized by its highest power of the variable being 2, takes the general form of ax^2 + bx + c = 0, where a, b, and c are constants, and a is not equal to zero. Tackling quadratic equations head-on requires a deep understanding of various solution methods, each offering a unique pathway to unravel the unknown values of x that satisfy the equation.
In this comprehensive guide, we embark on a journey to explore the intricacies of solving the quadratic equation -3x^2 + 6x = -2. We'll delve into the core concepts, unravel the methods at our disposal, and meticulously walk through the steps to arrive at the solutions. Our focus will be on employing the quadratic formula, a powerful and versatile technique that provides a guaranteed path to the roots of any quadratic equation. By the end of this exploration, you'll be equipped with the knowledge and skills to confidently tackle a wide range of quadratic equations.
Unveiling the Quadratic Formula: A Universal Solver
The quadratic formula, a cornerstone of algebra, emerges as a reliable and universally applicable method for finding the solutions to any quadratic equation. It provides a direct pathway to the roots, regardless of the complexity of the coefficients or the nature of the solutions. The formula elegantly expresses the solutions, denoted as x, in terms of the coefficients a, b, and c of the quadratic equation ax^2 + bx + c = 0:
x = (-b ± √(b^2 - 4ac)) / (2a)
Within this formula lies the discriminant, a crucial component that unveils the nature of the solutions. The discriminant, represented as b^2 - 4ac, holds the key to understanding whether the quadratic equation has two distinct real solutions, one repeated real solution, or two complex solutions.
- If b^2 - 4ac > 0: The equation possesses two distinct real solutions.
- If b^2 - 4ac = 0: The equation has one repeated real solution.
- If b^2 - 4ac < 0: The equation has two complex solutions.
Rewriting the Equation: Preparing for the Formula
Before we can unleash the power of the quadratic formula, we must first ensure that our equation, -3x^2 + 6x = -2, is in the standard quadratic form of ax^2 + bx + c = 0. This involves rearranging the terms to bring all elements to one side of the equation, leaving zero on the other side. To achieve this, we add 2 to both sides of the equation:
-3x^2 + 6x + 2 = 0
Now, we have successfully transformed the equation into the standard form, making it ready for the application of the quadratic formula.
Identifying Coefficients: Extracting the Key Values
With the equation in standard form, the next step is to identify the coefficients a, b, and c. These coefficients are the numerical values that multiply the respective terms in the quadratic equation. By carefully examining our equation, -3x^2 + 6x + 2 = 0, we can extract the following coefficients:
- a = -3
- b = 6
- c = 2
These coefficients will serve as the building blocks for our journey through the quadratic formula.
Plugging into the Formula: The Substitution Stage
Now comes the crucial step of substituting the identified coefficients into the quadratic formula. This is where we replace the symbolic representations of a, b, and c with their numerical counterparts. The quadratic formula, as we recall, is:
x = (-b ± √(b^2 - 4ac)) / (2a)
Substituting the values a = -3, b = 6, and c = 2 into the formula, we get:
x = (-6 ± √(6^2 - 4 * -3 * 2)) / (2 * -3)
This substitution sets the stage for the simplification process that will lead us to the solutions.
Simplifying the Expression: Unveiling the Solutions
The next phase involves simplifying the expression we obtained after substitution. This requires meticulous attention to the order of operations and careful execution of mathematical calculations. Let's break down the simplification process step by step:
-
Simplify the expression under the square root:
6^2 - 4 * -3 * 2 = 36 + 24 = 60
-
Substitute the simplified value back into the formula:
x = (-6 ± √60) / (-6)
-
Simplify the square root:
√60 = √(4 * 15) = 2√15
-
Substitute the simplified square root back into the formula:
x = (-6 ± 2√15) / (-6)
-
Divide both the numerator and denominator by -2:
x = (3 ± -√15) / 3
The Solutions: Unveiling the Roots
After the meticulous simplification process, we arrive at the solutions to the quadratic equation -3x^2 + 6x = -2. These solutions represent the values of x that satisfy the equation. From our simplified expression, we can extract two distinct solutions:
- x = (3 + √15) / 3
- x = (3 - √15) / 3
These are the roots of the equation, the points where the quadratic function intersects the x-axis. These solutions can also be expressed in a combined form:
x = (3 ± √15) / 3
This concise representation encapsulates both solutions in a single expression.
Conclusion: Mastering the Art of Solving Quadratic Equations
In this comprehensive guide, we embarked on a journey to solve the quadratic equation -3x^2 + 6x = -2. We meticulously explored the power of the quadratic formula, a universal tool for finding the solutions to any quadratic equation. We learned how to rewrite the equation in standard form, identify coefficients, substitute values into the formula, and simplify the resulting expression. Through this process, we unveiled the two distinct solutions to the equation:
- x = (3 + √15) / 3
- x = (3 - √15) / 3
By mastering the techniques presented in this guide, you are now equipped to confidently tackle a wide range of quadratic equations. Remember, practice is key to solidifying your understanding and honing your problem-solving skills. So, continue to explore, experiment, and embrace the challenges that quadratic equations present. With dedication and perseverance, you'll unlock the power of algebra and its ability to model and solve real-world problems.
Solving quadratic equations can sometimes feel like navigating a maze, but with the right tools, it becomes a much smoother journey. In this article, we'll explore the process of solving the quadratic equation -3x^2 + 6x = -2 using the quadratic formula, a reliable method for finding solutions to any quadratic equation. We'll break down each step, ensuring clarity and understanding along the way.
What is the Quadratic Formula?
The quadratic formula is a powerful tool used to find the solutions (also called roots) of a quadratic equation. A quadratic equation is an equation of the form ax^2 + bx + c = 0, where a, b, and c are constants, and a is not equal to zero. The quadratic formula is given by:
x = (-b ± √(b^2 - 4ac)) / (2a)
This formula provides two possible solutions for x, represented by the ± symbol. The expression inside the square root, b^2 - 4ac, is called the discriminant, and it tells us about the nature of the solutions:
- If b^2 - 4ac > 0, there are two distinct real solutions.
- If b^2 - 4ac = 0, there is exactly one real solution (a repeated root).
- If b^2 - 4ac < 0, there are two complex solutions.
Step-by-Step Solution for -3x^2 + 6x = -2
Let's now apply the quadratic formula to solve the equation -3x^2 + 6x = -2.
Step 1: Rewrite the equation in standard form
The first step is to rewrite the equation in the standard quadratic form, which is ax^2 + bx + c = 0. To do this, we need to move all terms to one side of the equation:
-3x^2 + 6x = -2
Add 2 to both sides:
-3x^2 + 6x + 2 = 0
Now, the equation is in the standard form, with a = -3, b = 6, and c = 2.
Step 2: Identify the coefficients
Next, we identify the coefficients a, b, and c from the standard form of the equation:
- a = -3
- b = 6
- c = 2
These coefficients will be used in the quadratic formula.
Step 3: Apply the quadratic formula
Now, we substitute the values of a, b, and c into the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
Substitute a = -3, b = 6, and c = 2:
x = (-6 ± √(6^2 - 4(-3)(2))) / (2(-3))
Step 4: Simplify the expression
Now, we simplify the expression step by step:
x = (-6 ± √(36 + 24)) / (-6)
x = (-6 ± √60) / (-6)
We can simplify √60 by factoring out the largest perfect square, which is 4:
√60 = √(4 * 15) = 2√15
So, the expression becomes:
x = (-6 ± 2√15) / (-6)
Step 5: Further simplification
We can simplify further by dividing both the numerator and the denominator by their greatest common divisor, which is 2:
x = (-3 ± √15) / (-3)
To make the expression more readable, we can multiply both the numerator and the denominator by -1:
x = (3 ± √15) / 3
Step 6: Final solutions
The quadratic formula gives us two solutions:
x1 = (3 + √15) / 3
x2 = (3 - √15) / 3
These are the two values of x that satisfy the original equation -3x^2 + 6x = -2.
Conclusion: Mastering the Quadratic Formula
In this article, we walked through the process of solving the quadratic equation -3x^2 + 6x = -2 using the quadratic formula. By rewriting the equation in standard form, identifying the coefficients, substituting them into the formula, and simplifying the expression, we found the two solutions:
- x1 = (3 + √15) / 3
- x2 = (3 - √15) / 3
The quadratic formula is a versatile tool that can be used to solve any quadratic equation, making it an essential technique for anyone studying algebra and beyond. Practice using the formula with different equations to build your confidence and skills.
In this article, we are going to understand the solutions for the quadratic equation -3x^2 + 6x = -2. Solving equations, especially quadratic ones, is a fundamental concept in algebra and has numerous applications in various fields, including physics, engineering, and economics. In this guide, we'll explore the steps to solve the given quadratic equation, interpret the solutions, and understand their significance.
What is a Quadratic Equation?
A quadratic equation is a polynomial equation of the second degree. The general form of a quadratic equation is:
ax^2 + bx + c = 0
where a, b, and c are constants, and a is not equal to zero. The solutions to a quadratic equation are the values of x that satisfy the equation, and they are also known as roots or zeros of the quadratic function. Quadratic equations can have two real solutions, one real solution (a repeated root), or two complex solutions, depending on the discriminant (b^2 - 4ac).
Solving -3x^2 + 6x = -2
To solve the quadratic equation -3x^2 + 6x = -2, we can use the quadratic formula, which is a reliable method for finding solutions to any quadratic equation. The quadratic formula is given by:
x = (-b ± √(b^2 - 4ac)) / (2a)
Let's break down the steps to solve the equation.
Step 1: Rewrite the equation in standard form
The first step is to rewrite the equation in the standard quadratic form, which is ax^2 + bx + c = 0. To do this, we need to move all terms to one side of the equation:
-3x^2 + 6x = -2
Add 2 to both sides:
-3x^2 + 6x + 2 = 0
Now, the equation is in the standard form.
Step 2: Identify the coefficients
Next, we identify the coefficients a, b, and c from the standard form of the equation:
- a = -3
- b = 6
- c = 2
These coefficients will be used in the quadratic formula.
Step 3: Apply the quadratic formula
Now, we substitute the values of a, b, and c into the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
Substitute a = -3, b = 6, and c = 2:
x = (-6 ± √(6^2 - 4(-3)(2))) / (2(-3))
Step 4: Simplify the expression
Now, we simplify the expression step by step:
x = (-6 ± √(36 + 24)) / (-6)
x = (-6 ± √60) / (-6)
We can simplify √60 by factoring out the largest perfect square, which is 4:
√60 = √(4 * 15) = 2√15
So, the expression becomes:
x = (-6 ± 2√15) / (-6)
Step 5: Further simplification
We can simplify further by dividing both the numerator and the denominator by their greatest common divisor, which is 2:
x = (-3 ± √15) / (-3)
To make the expression more readable, we can multiply both the numerator and the denominator by -1:
x = (3 ± √15) / 3
Step 6: Final solutions
The quadratic formula gives us two solutions:
x1 = (3 + √15) / 3
x2 = (3 - √15) / 3
These are the two values of x that satisfy the original equation -3x^2 + 6x = -2.
Interpreting the Solutions
The solutions to the quadratic equation -3x^2 + 6x = -2 are x1 = (3 + √15) / 3 and x2 = (3 - √15) / 3. These are real numbers, which means the parabola represented by the equation intersects the x-axis at two distinct points. The solutions represent the x-coordinates of these intersection points.
In decimal form, the solutions are approximately:
x1 ≈ (3 + 3.873) / 3 ≈ 2.291
x2 ≈ (3 - 3.873) / 3 ≈ -0.291
These approximate values can be useful for graphing the quadratic function and visualizing the solutions.
Conclusion: Understanding the Solutions of Quadratic Equations
In this article, we solved the quadratic equation -3x^2 + 6x = -2 using the quadratic formula. We rewrote the equation in standard form, identified the coefficients, applied the formula, and simplified the expression to find the two solutions:
- x1 = (3 + √15) / 3
- x2 = (3 - √15) / 3
These solutions represent the values of x that satisfy the equation and are the x-coordinates of the points where the quadratic function intersects the x-axis. Understanding how to solve quadratic equations and interpret their solutions is crucial in mathematics and many applied fields. By practicing these steps and exploring different equations, you can build your problem-solving skills and deepen your understanding of algebra.